LOCAL FIELDS AND TREES
7
and attains the same values; that is the set { d(x, y) : x, y
E
X} is the same for all
distances defined as above, independently of the choice of the origin o.
These observations prove that every element of
Aut(X)
acts continuously on
n.
This is clear for every rotation because a rotation is an isometry with respect to
an equivalent metric. Since every automorphism can be written as the product of
rotations and a single translation of step one, it is sufficient to show that, with
respect to the metric defined by fixing a vertex, say o, a step one translation along
a geodesic containing o is continuous. This is easily seen with a simple geometric
argument (a ball is either contracted in a smaller ball or dilated into a finite union
of balls).
THE TREE OF A LOCAL FIELD
We refer to [7] for definitions and basic properties of a local field. We shall also
follow if not otherwise stated the symbols and terminology of [7]. Thus we shall
denote a local field by F. We recall here that on F is defined the modular function
!alp
which satisfies the multiplicative property
and the ultrametric inequality
The modular function is nonnegative and attains the value zero only at 0.
It
defines therefore a metric on F. The ultrametric property implies that closed balls
of a given radius form a partition of F. We shall presently describe these balls.
Recall that the unit ball
0
={a
E
F:
!alp~
1}
is compact and open.
The ultrametric inequality implies that 0 is a ring. As such it contains a unique
maximal ideal, namely
P
={a
E
F:
!alp
1}.
We shall also use the hotation
ox ={a
E F:
!alp=
1},
so that
0
=
P
U
0
x .
Let p be a generator of P (we depart in this case from the notation of [7], where
the symbol
1r
is used in place of p). Then !PIP
= l,
where
q
is the order of the
q
finite field 0 /P, and P
=
pO.
The compact open sets
form a basis of neighborhoods of 0 which
fork-+
-oo exaust
F.
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